# Properties

 Label 1368.4.a.a.1.1 Level $1368$ Weight $4$ Character 1368.1 Self dual yes Analytic conductor $80.715$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1368.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$80.7146128879$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 1368.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.27492 q^{5} +12.0997 q^{7} +O(q^{10})$$ $$q-1.27492 q^{5} +12.0997 q^{7} -18.9244 q^{11} -55.9244 q^{13} +89.7492 q^{17} -19.0000 q^{19} +135.072 q^{23} -123.375 q^{25} +102.474 q^{29} -103.698 q^{31} -15.4261 q^{35} +29.6977 q^{37} -234.743 q^{41} -53.3231 q^{43} +33.3713 q^{47} -196.598 q^{49} -93.1752 q^{53} +24.1271 q^{55} -637.320 q^{59} -125.571 q^{61} +71.2990 q^{65} +119.375 q^{67} -18.5083 q^{71} -394.794 q^{73} -228.979 q^{77} +303.341 q^{79} +394.337 q^{83} -114.423 q^{85} +1021.88 q^{89} -676.667 q^{91} +24.2234 q^{95} +1322.82 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{5} - 6q^{7} + O(q^{10})$$ $$2q + 5q^{5} - 6q^{7} + 15q^{11} - 59q^{13} + 104q^{17} - 38q^{19} + 21q^{23} - 209q^{25} + 137q^{29} + 4q^{31} - 129q^{35} - 152q^{37} + 210q^{41} + 67q^{43} - 273q^{47} - 212q^{49} - 209q^{53} + 237q^{55} - 799q^{59} + 149q^{61} + 52q^{65} + 201q^{67} - 792q^{71} - 246q^{73} - 843q^{77} - 254q^{79} - 374q^{83} - 25q^{85} + 564q^{89} - 621q^{91} - 95q^{95} - 178q^{97} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.27492 −0.114032 −0.0570160 0.998373i $$-0.518159\pi$$
−0.0570160 + 0.998373i $$0.518159\pi$$
$$6$$ 0 0
$$7$$ 12.0997 0.653321 0.326660 0.945142i $$-0.394077\pi$$
0.326660 + 0.945142i $$0.394077\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −18.9244 −0.518721 −0.259360 0.965781i $$-0.583512\pi$$
−0.259360 + 0.965781i $$0.583512\pi$$
$$12$$ 0 0
$$13$$ −55.9244 −1.19313 −0.596563 0.802566i $$-0.703468\pi$$
−0.596563 + 0.802566i $$0.703468\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 89.7492 1.28043 0.640217 0.768194i $$-0.278845\pi$$
0.640217 + 0.768194i $$0.278845\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 135.072 1.22454 0.612272 0.790647i $$-0.290256\pi$$
0.612272 + 0.790647i $$0.290256\pi$$
$$24$$ 0 0
$$25$$ −123.375 −0.986997
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 102.474 0.656172 0.328086 0.944648i $$-0.393596\pi$$
0.328086 + 0.944648i $$0.393596\pi$$
$$30$$ 0 0
$$31$$ −103.698 −0.600795 −0.300398 0.953814i $$-0.597119\pi$$
−0.300398 + 0.953814i $$0.597119\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −15.4261 −0.0744995
$$36$$ 0 0
$$37$$ 29.6977 0.131953 0.0659766 0.997821i $$-0.478984\pi$$
0.0659766 + 0.997821i $$0.478984\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −234.743 −0.894162 −0.447081 0.894494i $$-0.647536\pi$$
−0.447081 + 0.894494i $$0.647536\pi$$
$$42$$ 0 0
$$43$$ −53.3231 −0.189109 −0.0945546 0.995520i $$-0.530143\pi$$
−0.0945546 + 0.995520i $$0.530143\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 33.3713 0.103568 0.0517841 0.998658i $$-0.483509\pi$$
0.0517841 + 0.998658i $$0.483509\pi$$
$$48$$ 0 0
$$49$$ −196.598 −0.573172
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −93.1752 −0.241483 −0.120742 0.992684i $$-0.538527\pi$$
−0.120742 + 0.992684i $$0.538527\pi$$
$$54$$ 0 0
$$55$$ 24.1271 0.0591508
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −637.320 −1.40630 −0.703152 0.711039i $$-0.748225\pi$$
−0.703152 + 0.711039i $$0.748225\pi$$
$$60$$ 0 0
$$61$$ −125.571 −0.263568 −0.131784 0.991278i $$-0.542071\pi$$
−0.131784 + 0.991278i $$0.542071\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 71.2990 0.136055
$$66$$ 0 0
$$67$$ 119.375 0.217671 0.108835 0.994060i $$-0.465288\pi$$
0.108835 + 0.994060i $$0.465288\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −18.5083 −0.0309370 −0.0154685 0.999880i $$-0.504924\pi$$
−0.0154685 + 0.999880i $$0.504924\pi$$
$$72$$ 0 0
$$73$$ −394.794 −0.632975 −0.316487 0.948597i $$-0.602503\pi$$
−0.316487 + 0.948597i $$0.602503\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −228.979 −0.338891
$$78$$ 0 0
$$79$$ 303.341 0.432006 0.216003 0.976393i $$-0.430698\pi$$
0.216003 + 0.976393i $$0.430698\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 394.337 0.521496 0.260748 0.965407i $$-0.416031\pi$$
0.260748 + 0.965407i $$0.416031\pi$$
$$84$$ 0 0
$$85$$ −114.423 −0.146010
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1021.88 1.21707 0.608536 0.793526i $$-0.291757\pi$$
0.608536 + 0.793526i $$0.291757\pi$$
$$90$$ 0 0
$$91$$ −676.667 −0.779494
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 24.2234 0.0261607
$$96$$ 0 0
$$97$$ 1322.82 1.38466 0.692330 0.721581i $$-0.256584\pi$$
0.692330 + 0.721581i $$0.256584\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1923.06 −1.89457 −0.947285 0.320391i $$-0.896186\pi$$
−0.947285 + 0.320391i $$0.896186\pi$$
$$102$$ 0 0
$$103$$ 467.774 0.447487 0.223743 0.974648i $$-0.428172\pi$$
0.223743 + 0.974648i $$0.428172\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −260.468 −0.235330 −0.117665 0.993053i $$-0.537541\pi$$
−0.117665 + 0.993053i $$0.537541\pi$$
$$108$$ 0 0
$$109$$ −511.856 −0.449789 −0.224894 0.974383i $$-0.572204\pi$$
−0.224894 + 0.974383i $$0.572204\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1453.48 1.21002 0.605008 0.796220i $$-0.293170\pi$$
0.605008 + 0.796220i $$0.293170\pi$$
$$114$$ 0 0
$$115$$ −172.206 −0.139637
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1085.94 0.836534
$$120$$ 0 0
$$121$$ −972.866 −0.730929
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 316.657 0.226581
$$126$$ 0 0
$$127$$ −2166.29 −1.51360 −0.756799 0.653647i $$-0.773238\pi$$
−0.756799 + 0.653647i $$0.773238\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −329.310 −0.219633 −0.109817 0.993952i $$-0.535026\pi$$
−0.109817 + 0.993952i $$0.535026\pi$$
$$132$$ 0 0
$$133$$ −229.894 −0.149882
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −736.919 −0.459556 −0.229778 0.973243i $$-0.573800\pi$$
−0.229778 + 0.973243i $$0.573800\pi$$
$$138$$ 0 0
$$139$$ −3041.10 −1.85571 −0.927853 0.372947i $$-0.878347\pi$$
−0.927853 + 0.372947i $$0.878347\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1058.34 0.618899
$$144$$ 0 0
$$145$$ −130.646 −0.0748247
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2156.31 −1.18558 −0.592790 0.805357i $$-0.701974\pi$$
−0.592790 + 0.805357i $$0.701974\pi$$
$$150$$ 0 0
$$151$$ −1816.60 −0.979024 −0.489512 0.871997i $$-0.662825\pi$$
−0.489512 + 0.871997i $$0.662825\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 132.206 0.0685099
$$156$$ 0 0
$$157$$ −1118.10 −0.568368 −0.284184 0.958770i $$-0.591723\pi$$
−0.284184 + 0.958770i $$0.591723\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1634.33 0.800020
$$162$$ 0 0
$$163$$ −3304.95 −1.58812 −0.794060 0.607839i $$-0.792037\pi$$
−0.794060 + 0.607839i $$0.792037\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1750.74 −0.811237 −0.405618 0.914043i $$-0.632944\pi$$
−0.405618 + 0.914043i $$0.632944\pi$$
$$168$$ 0 0
$$169$$ 930.541 0.423551
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2698.18 −1.18577 −0.592887 0.805286i $$-0.702012\pi$$
−0.592887 + 0.805286i $$0.702012\pi$$
$$174$$ 0 0
$$175$$ −1492.79 −0.644825
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3867.80 −1.61505 −0.807523 0.589836i $$-0.799192\pi$$
−0.807523 + 0.589836i $$0.799192\pi$$
$$180$$ 0 0
$$181$$ 3858.74 1.58463 0.792315 0.610112i $$-0.208875\pi$$
0.792315 + 0.610112i $$0.208875\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −37.8621 −0.0150469
$$186$$ 0 0
$$187$$ −1698.45 −0.664187
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4111.06 −1.55741 −0.778706 0.627389i $$-0.784124\pi$$
−0.778706 + 0.627389i $$0.784124\pi$$
$$192$$ 0 0
$$193$$ −1825.36 −0.680789 −0.340395 0.940283i $$-0.610561\pi$$
−0.340395 + 0.940283i $$0.610561\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 767.088 0.277425 0.138713 0.990333i $$-0.455704\pi$$
0.138713 + 0.990333i $$0.455704\pi$$
$$198$$ 0 0
$$199$$ 3176.43 1.13151 0.565757 0.824572i $$-0.308584\pi$$
0.565757 + 0.824572i $$0.308584\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1239.90 0.428691
$$204$$ 0 0
$$205$$ 299.277 0.101963
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 359.564 0.119003
$$210$$ 0 0
$$211$$ 2460.54 0.802797 0.401399 0.915903i $$-0.368524\pi$$
0.401399 + 0.915903i $$0.368524\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 67.9825 0.0215645
$$216$$ 0 0
$$217$$ −1254.71 −0.392512
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5019.17 −1.52772
$$222$$ 0 0
$$223$$ 3731.73 1.12061 0.560304 0.828287i $$-0.310684\pi$$
0.560304 + 0.828287i $$0.310684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −653.188 −0.190985 −0.0954926 0.995430i $$-0.530443\pi$$
−0.0954926 + 0.995430i $$0.530443\pi$$
$$228$$ 0 0
$$229$$ 340.511 0.0982602 0.0491301 0.998792i $$-0.484355\pi$$
0.0491301 + 0.998792i $$0.484355\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3936.99 1.10696 0.553478 0.832864i $$-0.313300\pi$$
0.553478 + 0.832864i $$0.313300\pi$$
$$234$$ 0 0
$$235$$ −42.5456 −0.0118101
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4762.18 −1.28887 −0.644434 0.764660i $$-0.722907\pi$$
−0.644434 + 0.764660i $$0.722907\pi$$
$$240$$ 0 0
$$241$$ −3893.55 −1.04069 −0.520343 0.853957i $$-0.674196\pi$$
−0.520343 + 0.853957i $$0.674196\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 250.646 0.0653600
$$246$$ 0 0
$$247$$ 1062.56 0.273722
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1384.13 −0.348069 −0.174034 0.984740i $$-0.555680\pi$$
−0.174034 + 0.984740i $$0.555680\pi$$
$$252$$ 0 0
$$253$$ −2556.16 −0.635196
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4645.01 1.12742 0.563711 0.825972i $$-0.309373\pi$$
0.563711 + 0.825972i $$0.309373\pi$$
$$258$$ 0 0
$$259$$ 359.332 0.0862078
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2151.41 0.504416 0.252208 0.967673i $$-0.418843\pi$$
0.252208 + 0.967673i $$0.418843\pi$$
$$264$$ 0 0
$$265$$ 118.791 0.0275368
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 5768.66 1.30752 0.653758 0.756704i $$-0.273192\pi$$
0.653758 + 0.756704i $$0.273192\pi$$
$$270$$ 0 0
$$271$$ −6859.23 −1.53752 −0.768761 0.639537i $$-0.779126\pi$$
−0.768761 + 0.639537i $$0.779126\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2334.79 0.511976
$$276$$ 0 0
$$277$$ 1237.24 0.268371 0.134185 0.990956i $$-0.457158\pi$$
0.134185 + 0.990956i $$0.457158\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4355.21 −0.924592 −0.462296 0.886726i $$-0.652974\pi$$
−0.462296 + 0.886726i $$0.652974\pi$$
$$282$$ 0 0
$$283$$ −3651.29 −0.766949 −0.383474 0.923551i $$-0.625273\pi$$
−0.383474 + 0.923551i $$0.625273\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2840.31 −0.584174
$$288$$ 0 0
$$289$$ 3141.91 0.639510
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3391.25 0.676174 0.338087 0.941115i $$-0.390220\pi$$
0.338087 + 0.941115i $$0.390220\pi$$
$$294$$ 0 0
$$295$$ 812.530 0.160364
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −7553.84 −1.46104
$$300$$ 0 0
$$301$$ −645.192 −0.123549
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 160.092 0.0300552
$$306$$ 0 0
$$307$$ 4343.35 0.807452 0.403726 0.914880i $$-0.367715\pi$$
0.403726 + 0.914880i $$0.367715\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5671.28 −1.03405 −0.517024 0.855971i $$-0.672960\pi$$
−0.517024 + 0.855971i $$0.672960\pi$$
$$312$$ 0 0
$$313$$ −9449.71 −1.70648 −0.853241 0.521516i $$-0.825367\pi$$
−0.853241 + 0.521516i $$0.825367\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5390.75 0.955125 0.477562 0.878598i $$-0.341520\pi$$
0.477562 + 0.878598i $$0.341520\pi$$
$$318$$ 0 0
$$319$$ −1939.27 −0.340370
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1705.23 −0.293752
$$324$$ 0 0
$$325$$ 6899.65 1.17761
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 403.781 0.0676632
$$330$$ 0 0
$$331$$ −9230.14 −1.53273 −0.766366 0.642404i $$-0.777937\pi$$
−0.766366 + 0.642404i $$0.777937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −152.193 −0.0248214
$$336$$ 0 0
$$337$$ −7815.90 −1.26338 −0.631690 0.775221i $$-0.717638\pi$$
−0.631690 + 0.775221i $$0.717638\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1962.42 0.311645
$$342$$ 0 0
$$343$$ −6528.96 −1.02779
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −680.076 −0.105211 −0.0526057 0.998615i $$-0.516753\pi$$
−0.0526057 + 0.998615i $$0.516753\pi$$
$$348$$ 0 0
$$349$$ 3641.72 0.558558 0.279279 0.960210i $$-0.409905\pi$$
0.279279 + 0.960210i $$0.409905\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7885.46 1.18895 0.594477 0.804113i $$-0.297359\pi$$
0.594477 + 0.804113i $$0.297359\pi$$
$$354$$ 0 0
$$355$$ 23.5965 0.00352781
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2862.17 0.420779 0.210389 0.977618i $$-0.432527\pi$$
0.210389 + 0.977618i $$0.432527\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 503.330 0.0721794
$$366$$ 0 0
$$367$$ −9783.29 −1.39151 −0.695754 0.718280i $$-0.744930\pi$$
−0.695754 + 0.718280i $$0.744930\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1127.39 −0.157766
$$372$$ 0 0
$$373$$ 6551.84 0.909495 0.454747 0.890621i $$-0.349730\pi$$
0.454747 + 0.890621i $$0.349730\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5730.81 −0.782896
$$378$$ 0 0
$$379$$ −2228.78 −0.302071 −0.151035 0.988528i $$-0.548261\pi$$
−0.151035 + 0.988528i $$0.548261\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5223.85 0.696935 0.348468 0.937321i $$-0.386702\pi$$
0.348468 + 0.937321i $$0.386702\pi$$
$$384$$ 0 0
$$385$$ 291.930 0.0386444
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 7672.26 0.999998 0.499999 0.866026i $$-0.333334\pi$$
0.499999 + 0.866026i $$0.333334\pi$$
$$390$$ 0 0
$$391$$ 12122.6 1.56795
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −386.734 −0.0492625
$$396$$ 0 0
$$397$$ −13564.0 −1.71475 −0.857375 0.514692i $$-0.827906\pi$$
−0.857375 + 0.514692i $$0.827906\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6488.39 0.808017 0.404009 0.914755i $$-0.367617\pi$$
0.404009 + 0.914755i $$0.367617\pi$$
$$402$$ 0 0
$$403$$ 5799.23 0.716825
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −562.011 −0.0684469
$$408$$ 0 0
$$409$$ 14696.2 1.77672 0.888360 0.459147i $$-0.151845\pi$$
0.888360 + 0.459147i $$0.151845\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −7711.36 −0.918768
$$414$$ 0 0
$$415$$ −502.747 −0.0594672
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −9492.08 −1.10673 −0.553363 0.832940i $$-0.686656\pi$$
−0.553363 + 0.832940i $$0.686656\pi$$
$$420$$ 0 0
$$421$$ −13028.4 −1.50823 −0.754116 0.656741i $$-0.771935\pi$$
−0.754116 + 0.656741i $$0.771935\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −11072.8 −1.26378
$$426$$ 0 0
$$427$$ −1519.36 −0.172195
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5404.99 0.604058 0.302029 0.953299i $$-0.402336\pi$$
0.302029 + 0.953299i $$0.402336\pi$$
$$432$$ 0 0
$$433$$ 16745.4 1.85850 0.929252 0.369448i $$-0.120453\pi$$
0.929252 + 0.369448i $$0.120453\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2566.37 −0.280930
$$438$$ 0 0
$$439$$ 13422.6 1.45928 0.729641 0.683831i $$-0.239687\pi$$
0.729641 + 0.683831i $$0.239687\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14048.7 1.50672 0.753358 0.657611i $$-0.228433\pi$$
0.753358 + 0.657611i $$0.228433\pi$$
$$444$$ 0 0
$$445$$ −1302.82 −0.138785
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9841.11 −1.03437 −0.517183 0.855875i $$-0.673020\pi$$
−0.517183 + 0.855875i $$0.673020\pi$$
$$450$$ 0 0
$$451$$ 4442.37 0.463820
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 862.694 0.0888873
$$456$$ 0 0
$$457$$ 4443.15 0.454796 0.227398 0.973802i $$-0.426978\pi$$
0.227398 + 0.973802i $$0.426978\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12288.1 1.24146 0.620728 0.784026i $$-0.286837\pi$$
0.620728 + 0.784026i $$0.286837\pi$$
$$462$$ 0 0
$$463$$ 4814.38 0.483246 0.241623 0.970370i $$-0.422320\pi$$
0.241623 + 0.970370i $$0.422320\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9863.03 −0.977316 −0.488658 0.872475i $$-0.662513\pi$$
−0.488658 + 0.872475i $$0.662513\pi$$
$$468$$ 0 0
$$469$$ 1444.39 0.142209
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1009.11 0.0980949
$$474$$ 0 0
$$475$$ 2344.12 0.226433
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −12743.3 −1.21557 −0.607785 0.794102i $$-0.707942\pi$$
−0.607785 + 0.794102i $$0.707942\pi$$
$$480$$ 0 0
$$481$$ −1660.83 −0.157437
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1686.48 −0.157896
$$486$$ 0 0
$$487$$ −4200.07 −0.390807 −0.195404 0.980723i $$-0.562602\pi$$
−0.195404 + 0.980723i $$0.562602\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 11292.3 1.03791 0.518955 0.854802i $$-0.326321\pi$$
0.518955 + 0.854802i $$0.326321\pi$$
$$492$$ 0 0
$$493$$ 9196.98 0.840185
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −223.944 −0.0202118
$$498$$ 0 0
$$499$$ −12126.6 −1.08790 −0.543948 0.839119i $$-0.683071\pi$$
−0.543948 + 0.839119i $$0.683071\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2033.61 0.180267 0.0901336 0.995930i $$-0.471271\pi$$
0.0901336 + 0.995930i $$0.471271\pi$$
$$504$$ 0 0
$$505$$ 2451.74 0.216042
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1367.41 0.119076 0.0595379 0.998226i $$-0.481037\pi$$
0.0595379 + 0.998226i $$0.481037\pi$$
$$510$$ 0 0
$$511$$ −4776.88 −0.413535
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −596.373 −0.0510279
$$516$$ 0 0
$$517$$ −631.532 −0.0537229
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 9613.98 0.808438 0.404219 0.914662i $$-0.367543\pi$$
0.404219 + 0.914662i $$0.367543\pi$$
$$522$$ 0 0
$$523$$ −8185.79 −0.684397 −0.342199 0.939628i $$-0.611172\pi$$
−0.342199 + 0.939628i $$0.611172\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9306.78 −0.769278
$$528$$ 0 0
$$529$$ 6077.52 0.499508
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 13127.8 1.06685
$$534$$ 0 0
$$535$$ 332.075 0.0268352
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3720.50 0.297316
$$540$$ 0 0
$$541$$ −1869.73 −0.148587 −0.0742937 0.997236i $$-0.523670\pi$$
−0.0742937 + 0.997236i $$0.523670\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 652.575 0.0512903
$$546$$ 0 0
$$547$$ 4024.62 0.314589 0.157295 0.987552i $$-0.449723\pi$$
0.157295 + 0.987552i $$0.449723\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1947.01 −0.150536
$$552$$ 0 0
$$553$$ 3670.32 0.282239
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15750.7 1.19816 0.599082 0.800688i $$-0.295532\pi$$
0.599082 + 0.800688i $$0.295532\pi$$
$$558$$ 0 0
$$559$$ 2982.06 0.225631
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 641.091 0.0479907 0.0239954 0.999712i $$-0.492361\pi$$
0.0239954 + 0.999712i $$0.492361\pi$$
$$564$$ 0 0
$$565$$ −1853.06 −0.137981
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18392.3 1.35509 0.677544 0.735482i $$-0.263044\pi$$
0.677544 + 0.735482i $$0.263044\pi$$
$$570$$ 0 0
$$571$$ 1400.26 0.102625 0.0513126 0.998683i $$-0.483660\pi$$
0.0513126 + 0.998683i $$0.483660\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −16664.5 −1.20862
$$576$$ 0 0
$$577$$ 23464.7 1.69298 0.846489 0.532406i $$-0.178712\pi$$
0.846489 + 0.532406i $$0.178712\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4771.35 0.340704
$$582$$ 0 0
$$583$$ 1763.29 0.125262
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1336.76 0.0939934 0.0469967 0.998895i $$-0.485035\pi$$
0.0469967 + 0.998895i $$0.485035\pi$$
$$588$$ 0 0
$$589$$ 1970.26 0.137832
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 14890.5 1.03116 0.515581 0.856841i $$-0.327576\pi$$
0.515581 + 0.856841i $$0.327576\pi$$
$$594$$ 0 0
$$595$$ −1384.48 −0.0953917
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −22648.7 −1.54491 −0.772456 0.635068i $$-0.780972\pi$$
−0.772456 + 0.635068i $$0.780972\pi$$
$$600$$ 0 0
$$601$$ 8930.64 0.606137 0.303069 0.952969i $$-0.401989\pi$$
0.303069 + 0.952969i $$0.401989\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1240.32 0.0833493
$$606$$ 0 0
$$607$$ −5611.75 −0.375245 −0.187623 0.982241i $$-0.560078\pi$$
−0.187623 + 0.982241i $$0.560078\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1866.27 −0.123570
$$612$$ 0 0
$$613$$ −16212.8 −1.06823 −0.534117 0.845410i $$-0.679356\pi$$
−0.534117 + 0.845410i $$0.679356\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26562.7 1.73318 0.866592 0.499017i $$-0.166306\pi$$
0.866592 + 0.499017i $$0.166306\pi$$
$$618$$ 0 0
$$619$$ 816.376 0.0530096 0.0265048 0.999649i $$-0.491562\pi$$
0.0265048 + 0.999649i $$0.491562\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12364.5 0.795139
$$624$$ 0 0
$$625$$ 15018.1 0.961159
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2665.34 0.168957
$$630$$ 0 0
$$631$$ 29218.6 1.84338 0.921690 0.387928i $$-0.126809\pi$$
0.921690 + 0.387928i $$0.126809\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2761.84 0.172599
$$636$$ 0 0
$$637$$ 10994.6 0.683867
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4831.98 0.297740 0.148870 0.988857i $$-0.452436\pi$$
0.148870 + 0.988857i $$0.452436\pi$$
$$642$$ 0 0
$$643$$ 25259.5 1.54920 0.774602 0.632449i $$-0.217950\pi$$
0.774602 + 0.632449i $$0.217950\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −11004.6 −0.668676 −0.334338 0.942453i $$-0.608513\pi$$
−0.334338 + 0.942453i $$0.608513\pi$$
$$648$$ 0 0
$$649$$ 12060.9 0.729479
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −5034.69 −0.301720 −0.150860 0.988555i $$-0.548204\pi$$
−0.150860 + 0.988555i $$0.548204\pi$$
$$654$$ 0 0
$$655$$ 419.843 0.0250452
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 5027.20 0.297165 0.148583 0.988900i $$-0.452529\pi$$
0.148583 + 0.988900i $$0.452529\pi$$
$$660$$ 0 0
$$661$$ −28126.0 −1.65503 −0.827515 0.561444i $$-0.810246\pi$$
−0.827515 + 0.561444i $$0.810246\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 293.095 0.0170914
$$666$$ 0 0
$$667$$ 13841.4 0.803512
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2376.35 0.136718
$$672$$ 0 0
$$673$$ 15864.9 0.908689 0.454344 0.890826i $$-0.349874\pi$$
0.454344 + 0.890826i $$0.349874\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −15911.8 −0.903308 −0.451654 0.892193i $$-0.649166\pi$$
−0.451654 + 0.892193i $$0.649166\pi$$
$$678$$ 0 0
$$679$$ 16005.7 0.904626
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 30172.0 1.69033 0.845167 0.534502i $$-0.179501\pi$$
0.845167 + 0.534502i $$0.179501\pi$$
$$684$$ 0 0
$$685$$ 939.510 0.0524042
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 5210.77 0.288120
$$690$$ 0 0
$$691$$ −14213.9 −0.782522 −0.391261 0.920280i $$-0.627961\pi$$
−0.391261 + 0.920280i $$0.627961\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3877.16 0.211610
$$696$$ 0 0
$$697$$ −21067.9 −1.14491
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −25215.2 −1.35858 −0.679292 0.733869i $$-0.737713\pi$$
−0.679292 + 0.733869i $$0.737713\pi$$
$$702$$ 0 0
$$703$$ −564.256 −0.0302721
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −23268.4 −1.23776
$$708$$ 0 0
$$709$$ 34384.4 1.82134 0.910672 0.413130i $$-0.135564\pi$$
0.910672 + 0.413130i $$0.135564\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −14006.7 −0.735700
$$714$$ 0 0
$$715$$ −1349.29 −0.0705744
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −4418.46 −0.229180 −0.114590 0.993413i $$-0.536555\pi$$
−0.114590 + 0.993413i $$0.536555\pi$$
$$720$$ 0 0
$$721$$ 5659.91 0.292353
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −12642.7 −0.647640
$$726$$ 0 0
$$727$$ 805.044 0.0410694 0.0205347 0.999789i $$-0.493463\pi$$
0.0205347 + 0.999789i $$0.493463\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4785.70 −0.242142
$$732$$ 0 0
$$733$$ −1662.37 −0.0837668 −0.0418834 0.999123i $$-0.513336\pi$$
−0.0418834 + 0.999123i $$0.513336\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2259.09 −0.112910
$$738$$ 0 0
$$739$$ 3145.59 0.156580 0.0782899 0.996931i $$-0.475054\pi$$
0.0782899 + 0.996931i $$0.475054\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −35469.5 −1.75134 −0.875672 0.482906i $$-0.839581\pi$$
−0.875672 + 0.482906i $$0.839581\pi$$
$$744$$ 0 0
$$745$$ 2749.11 0.135194
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −3151.57 −0.153746
$$750$$ 0 0
$$751$$ −29978.8 −1.45665 −0.728324 0.685233i $$-0.759700\pi$$
−0.728324 + 0.685233i $$0.759700\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 2316.01 0.111640
$$756$$ 0 0
$$757$$ −33882.6 −1.62680 −0.813398 0.581707i $$-0.802385\pi$$
−0.813398 + 0.581707i $$0.802385\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9023.05 −0.429810 −0.214905 0.976635i $$-0.568944\pi$$
−0.214905 + 0.976635i $$0.568944\pi$$
$$762$$ 0 0
$$763$$ −6193.29 −0.293856
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 35641.7 1.67790
$$768$$ 0 0
$$769$$ −773.311 −0.0362631 −0.0181315 0.999836i $$-0.505772\pi$$
−0.0181315 + 0.999836i $$0.505772\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −19778.7 −0.920299 −0.460150 0.887841i $$-0.652204\pi$$
−0.460150 + 0.887841i $$0.652204\pi$$
$$774$$ 0 0
$$775$$ 12793.7 0.592983
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4460.11 0.205135
$$780$$ 0 0
$$781$$ 350.258 0.0160477
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1425.48 0.0648122
$$786$$ 0 0
$$787$$ 32308.3 1.46336 0.731681 0.681648i $$-0.238736\pi$$
0.731681 + 0.681648i $$0.238736\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 17586.6 0.790528
$$792$$ 0 0
$$793$$ 7022.46 0.314470
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −40290.5 −1.79067 −0.895335 0.445394i $$-0.853064\pi$$
−0.895335 + 0.445394i $$0.853064\pi$$
$$798$$ 0 0
$$799$$ 2995.04 0.132612
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 7471.25 0.328337
$$804$$ 0 0
$$805$$ −2083.64 −0.0912279
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 16600.1 0.721417 0.360709 0.932679i $$-0.382535\pi$$
0.360709 + 0.932679i $$0.382535\pi$$
$$810$$ 0 0
$$811$$ 9586.41 0.415073 0.207537 0.978227i $$-0.433455\pi$$
0.207537 + 0.978227i $$0.433455\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 4213.54 0.181097
$$816$$ 0 0
$$817$$ 1013.14 0.0433846
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 33381.4 1.41902 0.709512 0.704693i $$-0.248915\pi$$
0.709512 + 0.704693i $$0.248915\pi$$
$$822$$ 0 0
$$823$$ −3953.76 −0.167460 −0.0837299 0.996488i $$-0.526683\pi$$
−0.0837299 + 0.996488i $$0.526683\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 33403.5 1.40454 0.702270 0.711910i $$-0.252170\pi$$
0.702270 + 0.711910i $$0.252170\pi$$
$$828$$ 0 0
$$829$$ 41612.3 1.74337 0.871686 0.490065i $$-0.163027\pi$$
0.871686 + 0.490065i $$0.163027\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −17644.5 −0.733909
$$834$$ 0 0
$$835$$ 2232.05 0.0925070
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −10843.5 −0.446199 −0.223099 0.974796i $$-0.571617\pi$$
−0.223099 + 0.974796i $$0.571617\pi$$
$$840$$ 0 0
$$841$$ −13888.0 −0.569438
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1186.36 −0.0482984
$$846$$ 0 0
$$847$$ −11771.4 −0.477531
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4011.33 0.161583
$$852$$ 0 0
$$853$$ 13530.7 0.543119 0.271560 0.962422i $$-0.412461\pi$$
0.271560 + 0.962422i $$0.412461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3510.58 0.139929 0.0699645 0.997549i $$-0.477711\pi$$
0.0699645 + 0.997549i $$0.477711\pi$$
$$858$$ 0 0
$$859$$ −4945.74 −0.196445 −0.0982226 0.995164i $$-0.531316\pi$$
−0.0982226 + 0.995164i $$0.531316\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 38957.6 1.53665 0.768326 0.640059i $$-0.221090\pi$$
0.768326 + 0.640059i $$0.221090\pi$$
$$864$$ 0 0
$$865$$ 3439.96 0.135216
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −5740.54 −0.224090
$$870$$ 0 0
$$871$$ −6675.95 −0.259708
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3831.45 0.148030
$$876$$ 0 0
$$877$$ 544.671 0.0209718 0.0104859 0.999945i $$-0.496662\pi$$
0.0104859 + 0.999945i $$0.496662\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 19771.1 0.756079 0.378040 0.925789i $$-0.376598\pi$$
0.378040 + 0.925789i $$0.376598\pi$$
$$882$$ 0 0
$$883$$ −12249.6 −0.466854 −0.233427 0.972374i $$-0.574994\pi$$
−0.233427 + 0.972374i $$0.574994\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 21817.3 0.825877 0.412938 0.910759i $$-0.364503\pi$$
0.412938 + 0.910759i $$0.364503\pi$$
$$888$$ 0 0
$$889$$ −26211.4 −0.988866
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −634.054 −0.0237602
$$894$$ 0 0
$$895$$ 4931.13 0.184167
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −10626.3 −0.394225
$$900$$ 0 0
$$901$$ −8362.40 −0.309203
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4919.58 −0.180699
$$906$$ 0 0
$$907$$ −35592.0 −1.30299 −0.651496 0.758652i $$-0.725858\pi$$
−0.651496 + 0.758652i $$0.725858\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −952.654 −0.0346464 −0.0173232 0.999850i $$-0.505514\pi$$
−0.0173232 + 0.999850i $$0.505514\pi$$
$$912$$ 0 0
$$913$$ −7462.60 −0.270511
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3984.54 −0.143491
$$918$$ 0 0
$$919$$ −42326.8 −1.51929 −0.759647 0.650335i $$-0.774628\pi$$
−0.759647 + 0.650335i $$0.774628\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1035.06 0.0369118
$$924$$ 0 0
$$925$$ −3663.94 −0.130237
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 27446.4 0.969309 0.484654 0.874706i $$-0.338945\pi$$
0.484654 + 0.874706i $$0.338945\pi$$
$$930$$ 0 0
$$931$$ 3735.36 0.131495
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2165.38 0.0757387
$$936$$ 0 0
$$937$$ 15560.3 0.542511 0.271256 0.962507i $$-0.412561\pi$$
0.271256 + 0.962507i $$0.412561\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −17725.4 −0.614060 −0.307030 0.951700i $$-0.599335\pi$$
−0.307030 + 0.951700i $$0.599335\pi$$
$$942$$ 0 0
$$943$$ −31707.2 −1.09494
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 588.382 0.0201899 0.0100950 0.999949i $$-0.496787\pi$$
0.0100950 + 0.999949i $$0.496787\pi$$
$$948$$ 0 0
$$949$$ 22078.6 0.755219
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −28644.2 −0.973638 −0.486819 0.873503i $$-0.661843\pi$$
−0.486819 + 0.873503i $$0.661843\pi$$
$$954$$ 0 0
$$955$$ 5241.26 0.177595
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −8916.47 −0.300238
$$960$$ 0 0
$$961$$ −19037.8 −0.639045
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 2327.18 0.0776318
$$966$$ 0 0
$$967$$ 44523.4 1.48064 0.740318 0.672257i $$-0.234675\pi$$
0.740318 + 0.672257i $$0.234675\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −18242.7 −0.602922 −0.301461 0.953478i $$-0.597474\pi$$
−0.301461 + 0.953478i $$0.597474\pi$$
$$972$$ 0 0
$$973$$ −36796.4 −1.21237
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −23434.6 −0.767390 −0.383695 0.923460i $$-0.625349\pi$$
−0.383695 + 0.923460i $$0.625349\pi$$
$$978$$ 0 0
$$979$$ −19338.6 −0.631321
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 59519.4 1.93121 0.965603 0.260022i $$-0.0837299\pi$$
0.965603 + 0.260022i $$0.0837299\pi$$
$$984$$ 0 0
$$985$$ −977.974 −0.0316354
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7202.47 −0.231573
$$990$$ 0 0
$$991$$ 8466.93 0.271404 0.135702 0.990750i $$-0.456671\pi$$
0.135702 + 0.990750i $$0.456671\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4049.69 −0.129029
$$996$$ 0 0
$$997$$ 27474.3 0.872738 0.436369 0.899768i $$-0.356264\pi$$
0.436369 + 0.899768i $$0.356264\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1368.4.a.a.1.1 2
3.2 odd 2 152.4.a.a.1.1 2
12.11 even 2 304.4.a.e.1.2 2
24.5 odd 2 1216.4.a.m.1.2 2
24.11 even 2 1216.4.a.k.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.a.1.1 2 3.2 odd 2
304.4.a.e.1.2 2 12.11 even 2
1216.4.a.k.1.1 2 24.11 even 2
1216.4.a.m.1.2 2 24.5 odd 2
1368.4.a.a.1.1 2 1.1 even 1 trivial